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The document describes two numerical methods for finding roots of functions: the Bisection method and the Newton-Raphson method. Each method includes a function definition with input parameters, default values for optional parameters, and a loop for iterating until a root is found or a stopping criterion is met. The Bisection method requires two initial guesses that bracket a root, while the Newton-Raphson method uses an initial guess and the derivative of the function.

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11 views2 pages

M Files

The document describes two numerical methods for finding roots of functions: the Bisection method and the Newton-Raphson method. Each method includes a function definition with input parameters, default values for optional parameters, and a loop for iterating until a root is found or a stopping criterion is met. The Bisection method requires two initial guesses that bracket a root, while the Newton-Raphson method uses an initial guess and the derivative of the function.

Uploaded by

alicliverpool
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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1.

Bisection method

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function root = bisection(func,xl,xu,es,maxit)


% root = bisection(func,xl,xu,es,maxit):
% uses bisection method to find the root of a function
% input:
% func = name of function
% xl, xu = lower and upper guesses
% es = (optional) stopping criterion (%)
% maxit = (optional) maximum allowable iterations
% output:
% root = real root

if func(xl)*func(xu)>0 %if guesses do not bracket a sign change


disp('no bracket') %display an error message
return %and terminate
end
% if necessary, assign default values
if nargin<5, maxit=50; end %if maxit blank set to 50
if nargin<4, es=0.001; end %if es blank set to 0.001

% bisection
iter = 0;
xr = xl;
while (1)
xrold = xr;
xr = (xl + xu)/2;
iter = iter + 1;
if xr ~= 0, ea = abs((xr - xrold)/xr) * 100; end
test = func(xl)*func(xr);
if test < 0
xu = xr;
elseif test > 0
xl = xr;
else
ea = 0;
end
if ea <= es | iter >= maxit, break, end
end
root = xr;

______________________________________________________
2. Newton Raphson method

______________________________________________________

function root = newtraph(func,dfunc,xr,es,maxit)


% root = newtraph(func,dfunc,xguess,es,maxit):
% uses Newton-Raphson method to find the root of a function
% input:
% func = name of function
% dfunc = name of derivative of function
% xguess = initial guess
% es = (optional) stopping criterion (%)
% maxit = (optional) maximum allowable iterations
% output:
% root = real root

% if necessary, assign default values


if nargin<5, maxit=50; end %if maxit blank set to 50
if nargin<4, es=0.001; end %if es blank set to 0.001

% Newton-Raphson
iter = 0;
while (1)
xrold = xr;
xr = xr - func(xr)/dfunc(xr);
iter = iter + 1;
if xr ~= 0, ea = abs((xr - xrold)/xr) * 100; end
if ea <= es | iter >= maxit, break, end
end
root = xr;

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